Nonlinear water waves: introduction and overview

Abstract

For more than two centuries progress in the study of water waves proved to be interdependent with innovative and deep developments in theoretical and experimental directions of investigation. In recent years, considerable progress has been achieved towards the understanding of waves of large amplitude. Within this setting one cannot rely on linear theory as nonlinearity becomes an essential feature. Various analytic methods have been developed and adapted to come to terms with the challenges encountered in settings where approximations (such as those provided by linear or weakly nonlinear theory) are ineffective. Without relying on simpler models, progress becomes contingent upon the discovery of structural properties, the exploitation of which requires a combination of creative ideas and state-of-the-art technical tools. The successful quest for structure often reveals unexpected patterns and confers aesthetic value on some of these studies. The topics covered in this issue are both multi-disciplinary and interdisciplinary: there is a strong interplay between mathematical analysis, numerical computation and experimental/field data, interacting with each other via mutual stimulation and feedback. This theme issue reflects some of the new important developments that were discussed during the programme ‘Nonlinear water waves’ that took place at the Isaac Newton Institute for Mathematical Sciences (Cambridge, UK) from 31st July to 25th August 2017. A cross-section of the experts in the study of water waves who participated in the programme authored the collected papers. These papers illustrate the diversity, intensity and interconnectivity of the current research activity in this area. They offer new insight, present emerging theoretical methodologies and computational approaches, and describe sophisticated experimental results.

This article is part of the theme issue ‘Nonlinear water waves’.

The study of water waves has deep and fascinating connections to many scientific research areas, with the added advantage that very often the behaviour of the waves may be analysed by direct observation. The complexity and variety of water-wave phenomena require innovative tools from many research areas, ranging from physical approaches to abstract mathematical considerations. It seems to be generally accepted that the most promising directions of future research in water waves lie in the study of waves of large amplitude, where the classical linear approach offers little insight. Progress is contingent upon fruitful interdisciplinary endeavours between researchers with mathematical, physical and engineering expertise. The papers in this theme issue reflect the state of the art, covering theoretical, numerical and experimental aspects and thus opening new avenues for promising multi-disciplinary research. The 18 other articles in this theme issue cover a wide spectrum of research directions. While some of the papers can be approximately grouped according to their main topic, there are nevertheless overlaps between these categories as some phenomena that seem to be disparate turn out to have deep underlying interrelations.

It is widely accepted that the incompressible Euler equations with a free boundary are the governing equations for water waves and, as such, they have been the subject of a wide range of research. Most of the theoretical works on surface water waves assume that the flow is irrotational (zero vorticity). This is mathematically convenient because in this setting the flow velocity is the gradient of a velocity potential and thus harmonic function theory comes into play. Moreover, if at some instant the water flow is irrotational, it cannot acquire vorticity at later times unless non-conservative forces act. While the qualitative understanding of periodic travelling waves in irrotational flow over a flatbed is quite advanced, providing insight into the dynamics of waves of large amplitude [1–4], there are still important quantitative aspects that remain to be elucidated. Weakly nonlinear model equations, the most popular being integrable equations (because their rich structure permits, by means of inverse scattering approaches, a quite detailed understanding of their dynamics [5–7]), fail to address linear theory's inability to provide accurate quantitative predictions for waves of moderate and large amplitude. The papers [8–10] deal with the full nonlinearity of the governing equations for water waves, providing some valuable quantitative information. The contribution [11] discusses aspects of the propagation of irrotational water waves over a variable bottom.

While irrotational flows are physically relevant in some circumstances (e.g. for waves propagating into water previously at rest), many mechanisms of vorticity generation exist as vorticity is the hallmark of non-uniform underlying currents, which influence the form of the free surface to the extent that often they become a dominant feature of the wave dynamics (see the discussion in [5,12,13]). Some results for irrotational travelling waves have counterparts for rotational waves with no stagnation points: explicit dispersion relations were found for large classes of vorticity distributions [14], there is an existence theory for waves of large amplitude [15,16], regularity results hold (see the discussions in [17–19]) and one can also prove the somewhat unexpected feature that the monotonicity of the free surface between successive crests and troughs ensures symmetry despite the presence of an underlying sheared current [20]. However, non-zero vorticity brings about new phenomena, for example flow reversal and Kelvin's cat's-eye streamline patterns near a critical layer (see the discussions in [16,21]). Also, numerical simulations [22,23] indicate that, in contrast to irrotational travelling waves (which can always be represented by the graph of a function; see [24,25]), the interaction between travelling waves and adverse currents with constant non-zero vorticity results in bulbous waves with overhanging profiles. We point out that, for wave–current interactions, the assumption of constant vorticity simplifies the mathematics considerably and constant vorticity is representative when the waves are long compared with the water depth since in this case it is the existence of a non-zero mean vorticity that is important rather than its specific distribution [22]. The high-precision numerical simulations for travelling waves in flows with constant non-zero vorticity over a flatbed presented in [26,27] draw particular attention to the considerable differences that can exist between these flow patterns and the case of irrotational water–wave propagation.

Three papers in this issue are devoted to the study of the pressure beneath a surface wave. The reliance on pressure sensors for wave measurements is quite common, being in some circumstances the only practical system. A crucial aspect in this procedure is the necessity of relating the pressure data collected in depths (typically of up to 100 m) to the surface wave displacements. The framework of linear theory is used in practice but there are frequent discrepancies in excess of 15% between the predictions of linear wave theory and the actual wave heights (see the overview in [28] and the discussions in [29,30]). In order to address the inaccuracies of the linear theory, nonlinear effects have to be accounted for. Of interest is a better understanding of the behaviour of the pressure beneath the waves, as well as procedures on how to relate the variations of the pressure to the amplitude of the surface waves. For practical purposes it is convenient to regard the pressure beneath a surface wave as the sum of a dynamic part that encodes the fluid motion (related to the kinetic energy of the water flow) and a hydrostatic part due to the weight of the fluid. Recently, the locations of the extrema of the pressure and of the dynamic pressure beneath a periodic travelling wave in irrotational flow over a flatbed were determined without any restriction on the wave amplitude [31,32] and in [33] it was shown that the variation of the pressure on the bed, normalized by means of ρg, where ρ is the (constant) density and g is the gravitational constant of acceleration, provides a lower bound for the wave height. The mentioned results about the pressure in regular waves with no underlying current can be extended to allow for some underlying currents [34,35] but for irregular waves the state of the art is far from satisfactory (theoretically, as well as computationally). Consequently, the investigation of the pressure field beneath a surface water wave is currently of great interest. This topic is covered by three papers in this issue: theoretical aspects are addressed in [36,37], while [38] presents experimental studies.

Four papers in this issue are motivated by the recent identification of some non-trivial exact analytical solutions for the governing equations for geophysical flows, solutions that describe equatorial waves [39–41] or model some large-scale ocean currents with distinctive and persistent patterns—gyres [42], the ocean flow in the equatorial Pacific [43] and the ocean current surrounding Antarctica [44]. The papers [45,46] provide unified approaches to these exact solutions, while in [47] the issue of the stability of the equatorial and polar flows is addressed. A bifurcation approach to gain insight into equatorial ocean dynamics, where the presence of currents with flow reversal (see the data provided in [48]) leads to a rich variety of wave phenomena, is developed in [49].

Finally, a number of papers in this volume tackle individually emerging issues or bring new insight into some long-lasting questions related to water-wave phenomena, as follows.

• — Since the governing equations for water waves are highly nonlinear, solutions that describe realistic fluid motions are elusive and the development of singularities in classical solutions (in the form of wave breaking) is one of the most difficult unanswered questions. However, on the related issue of long-time existence of solutions of small amplitude there has recently been significant progress, discussed in [50].

• — The paper [51] is a broadly informative review of the history and present research directions associated with the phenomenon of Stokes drift by surface gravity waves.

• — Tsunamis are powerful ocean waves created by submarine earthquakes, land and ice slips, or meteor strikes. Unlike wind-generated waves, they often have wavelengths in excess of 100 km and periods of the order of 1 h and behave like shallow-water waves [52–55]. Given their typical wave characteristics, it is natural to ask whether tsunamis might be substantially altered by the Coriolis effect due to the Earth's rotation. This issue is addressed in [56].

• — Computations of solitary hydroelastic waves are presented in [57], using a recently derived nonlinear model for the interaction between a heavy thin elastic sheet and an infinite ocean beneath it [58]. The motivation is a better understanding of polar ice deformations subject to water-wave motions.

• — The paper [59] is devoted to a study of leading-order effects arising in the (statistical) description of random surface wave interaction on deep water.

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Competing interests

The author declares that he has no competing interests.

Funding

The support of the Isaac Newton Institute for Mathematical Sciences (Cambridge, UK) is gratefully acknowledged.